^{1,a)}, Moon S. Chung

^{2}, Peter B. Lerner

^{3}, Brock L. Weiss

^{4}, Nicholas M. Miskovsky

^{5}and Paul H. Cutler

^{5}

### Abstract

The authors simulate the rectification properties of geometrically asymmetric metal–vacuum–metal junctions in which one of the metals is flat while the other is extended by a sharp tip. The authors analyze, in particular, the efficiency with which the energy of incident radiations, with frequencies in the infrared through the visible, is transferred to the electrons that cross the junction. This time-dependent electronic scattering problem is solved by using a transfer-matrix methodology. In order to validate this technique, the results achieved by using this quantum-mechanical scheme are compared with those provided by models that are based on extrapolations of static current–voltage data. The authors then discuss concepts that are relevant to the efficiency with which energy is converted in these junctions. The authors finally analyze how this efficiency is affected by the amplitude and the angular frequency of the potentials that are induced in these junctions, the work function of the metallic contacts and the spacing between these contacts.

A.M. is funded as a Research Associate by the National Fund for Scientific Research (FNRS) of Belgium. This work used resources of the Interuniversity Scientific Computing Facility located at the University of Namur, Belgium, which is supported by the F.R.S.-FNRS under Convention No. 2.4617.07. The authors acknowledge the support of the Paul H. Cutler Endowment Fund for Excellence of the Eberly College of Science, The Pennsylvania State University. O. Deparis is acknowledged for his interest in this problem and for stimulating discussions.

I. INTRODUCTION

II. TIME-DEPENDENT MODELING OF ELECTRONIC SCATTERING USING A TRANSFER-MATRIX METHODOLOGY

III. MODELING TECHNIQUES BASED ON EXTRAPOLATIONS OF STATIC CURRENT−VOLTAGE DATA

A. Classical expressions for and

B. Finite-difference expressions for and

C. Tien-Gordon expressions for and

IV. COMPARISON BETWEEN THE DIFFERENT MODELS

V. QUANTUM EFFICIENCY AND ENERGY CONVERSION EFFICIENCY

VI. CONCLUSIONS

### Key Topics

- Photons
- 34.0
- Finite difference methods
- 27.0
- Work functions
- 15.0
- Rectification
- 11.0
- Energy transfer
- 9.0

## Figures

Static part (a) and oscillating part (b) of the potential energy that is used for the time-dependent transfer-matrix simulations. The gap spacing *D* is 2 nm. The radius of the hemispherical protrusion is 1 nm. The junction is subject to an electric potential , where *V* _{stat} = 0 V and *V* _{osc} = 0.1 V.

Static part (a) and oscillating part (b) of the potential energy that is used for the time-dependent transfer-matrix simulations. The gap spacing *D* is 2 nm. The radius of the hemispherical protrusion is 1 nm. The junction is subject to an electric potential , where *V* _{stat} = 0 V and *V* _{osc} = 0.1 V.

(Color online) Static *I* _{stat}(*V* _{stat}) data achieved for a gap spacing *D* of 2 nm and a work function *W* of 4.5 eV (solid line) and 1.5 eV (dashed line). *I* _{stat} is positive for *V* _{stat} > 0 V and negative for *V* _{stat} < 0 V. The temperature *T* is 300 K.

(Color online) Static *I* _{stat}(*V* _{stat}) data achieved for a gap spacing *D* of 2 nm and a work function *W* of 4.5 eV (solid line) and 1.5 eV (dashed line). *I* _{stat} is positive for *V* _{stat} > 0 V and negative for *V* _{stat} < 0 V. The temperature *T* is 300 K.

(Color online) Mean diode current provided by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line) and the time-dependent transfer-matrix technique (triangles). The representation also includes the results achieved using the transfer-matrix technique of Ref. 2 (crosses). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K.

(Color online) Mean diode current provided by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line) and the time-dependent transfer-matrix technique (triangles). The representation also includes the results achieved using the transfer-matrix technique of Ref. 2 (crosses). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K.

(Color online) Mean energy gained per unit of time by the electrons that cross the junction, as provided by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The representation also includes the results achieved using the transfer-matrix technique of Ref. 2 (crosses) as well as the result achieved from a classical integration of the transfer-matrix currents (squares). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K.

(Color online) Mean energy gained per unit of time by the electrons that cross the junction, as provided by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The representation also includes the results achieved using the transfer-matrix technique of Ref. 2 (crosses) as well as the result achieved from a classical integration of the transfer-matrix currents (squares). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K.

(Color online) Quantum efficiency obtained when calculating and by the classical expressions and (solid line), the finite-difference expressions and (dashed line), the Tien-Gordon expressions and (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K.

(Color online) Quantum efficiency obtained when calculating and by the classical expressions and (solid line), the finite-difference expressions and (dashed line), the Tien-Gordon expressions and (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K.

(Color online) Energy conversion efficiency obtained when calculating by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K. The vertical line indicates the height of the static part of the potential barrier for electrons at the Fermi level.

(Color online) Energy conversion efficiency obtained when calculating by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K. The vertical line indicates the height of the static part of the potential barrier for electrons at the Fermi level.

(Color online) Energy conversion efficiency obtained when calculating by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K. The vertical line indicates the height of the static part of the potential barrier for electrons at the Fermi level.

(Color online) Energy conversion efficiency obtained when calculating by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 1 V. The work function *W* is 4.5 eV. The temperature *T* is 300 K. The vertical line indicates the height of the static part of the potential barrier for electrons at the Fermi level.

(Color online) Energy conversion efficiency obtained when calculating by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V (a) and 1 V (b). The work function *W* is 1.5 eV. The temperature *T* is 300 K. The vertical lines indicate the height of the static part of the potential barrier for electrons at the Fermi level.

(Color online) Energy conversion efficiency obtained when calculating by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 2 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 0.1 V (a) and 1 V (b). The work function *W* is 1.5 eV. The temperature *T* is 300 K. The vertical lines indicate the height of the static part of the potential barrier for electrons at the Fermi level.

(Color online) Static *I* _{stat}(*V* _{stat}) data achieved for a gap spacing *D* of 1.5 nm and a work function *W* of 4.5 eV (solid line) and 1.5 eV (dashed line). *I* _{stat} is positive for *V* _{stat} > 0 V and negative for *V* _{stat} < 0 V. The temperature *T* is 300 K.

(Color online) Static *I* _{stat}(*V* _{stat}) data achieved for a gap spacing *D* of 1.5 nm and a work function *W* of 4.5 eV (solid line) and 1.5 eV (dashed line). *I* _{stat} is positive for *V* _{stat} > 0 V and negative for *V* _{stat} < 0 V. The temperature *T* is 300 K.

(Color online) Energy conversion efficiency obtained when calculating by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 1.5 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 1 V. The work function *W* is 4.5 eV (a) and 1.5 eV (b). The temperature *T* is 300 K. The vertical line indicates the height of the static part of the potential barrier for electrons at the Fermi level.

(Color online) Energy conversion efficiency obtained when calculating by the classical expression (solid line), the finite-difference expression (dashed line), the Tien-Gordon expression (dotted-dashed line), and the time-dependent transfer-matrix technique (triangles). The gap spacing *D* is 1.5 nm. The static voltage *V* _{stat} is 0 V. The amplitude *V* _{osc} of the oscillating voltage is 1 V. The work function *W* is 4.5 eV (a) and 1.5 eV (b). The temperature *T* is 300 K. The vertical line indicates the height of the static part of the potential barrier for electrons at the Fermi level.

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